This invention relates to an adaptive control device that is designed to identify a discrete-time transfer function model P(z) of a controlled object and generate a feedforward manipulated variable using the inverse model of P(z). This invention also relates to an image forming apparatus mounting the adaptive control device thereon, and a recording medium.
Conventionally, a known method of controlling a controlled object in accordance with an input target value improves following capability and response capability of the controlled object by combining feedforward control and feedback control as shown in FIG. 9.
The control system shown in FIG. 9 is a double-degree-of-freedom system including a feedforward controller 101 with a transfer function model CFF and a feedback controller 102 with a transfer function model CFB. The feedforward controller 101 generates and outputs a feedforward manipulated variable uFF to an input target value r. A subtractor 104 is provided which calculates a difference between the target value r and a control output value (controlled variable) y. The result e of calculation from the subtractor 104 is supplied to the feedback controller 102. Thereby, the feedback controller 102 generates and outputs a feedback manipulated variable uFB. The manipulated variables uFF and uFB are added in an adder 105. The result of addition is supplied to a controlled object 103 as a manipulated variable u.
In the above structured control system, the transfer function model CFF of the feedforward controller 101 is expressed by the inverse model 1/P of the controlled object 103. Accordingly, in order to effectively operate this control system, it is necessary that the transfer function model CFF of the feedforward controller 101 sufficiently matches a transfer function model of the controlled object 103.
However, even if the overall system is structured by designing the feedforward controller 101 using a transfer function model which sufficiently matches actual characteristics of the controlled object 103 at design time, the characteristics of the controlled object 103 may be changed from those at the design time owing to various factors such as aging and environmental change. The greater the change is, that is, the larger the gap is between the characteristics of the controlled object 103 set in the feedforward controller 101 and the actual characteristics of the controlled object 103, the larger feedback manipulated variable uFB is generated and outputted from the feedback controller 102 in order to fill up the gap. This affects the following capability and response capability which are the advantages of the feedforward control. Degradation is caused in control performance.
To solve the above problem, a known adaptive control technique adds an adaptive identifier which adaptively performs model identification of the controlled object. The model identification is performed online and the result of identification is reflected in the feedforward control system.
FIG. 10 shows an adaptive control system which combines adaptive control into the control system of FIG. 9. The adaptive control system of FIG. 10 includes an adaptive identifier 107 added to the control system of FIG. 9. The feedforward controller 110 is designed to update the transfer function model CFF based on the result of identification by the adaptive identifier 107. Change, if any, in actual characteristics of the controlled object 103 is reflected in the feedforward controller 110. Use of the adaptive control technique may maintain optimum control performance at all times.
The adaptive identifier 107 estimates each parameter (coefficient) of a transfer function model P of the controlled object 103 to identify the controlled object 103. FIG. 11 shows the schematic structure of the adaptive identifier 107. As shown in FIG. 11, the adaptive identifier 107 includes a model identifier 111 and a subtractor 112. In FIG. 11, a transfer function model P(z) of the controlled object 103 and a transfer function model {circumflex over (P)}(z) of the model identifier 111 (hereinafter, also referred to as “identification model”) are both discrete-time transfer functions. Also, k indicates time (timing).
In the adaptive identifier 107, the model identifier 111 identifies the controlled object 103 to obtain the identification model {circumflex over (P)}(z). That is, the model identifier 111 estimates each parameter of the transfer function model P(z) of the controlled object 103 to obtain the identification model {circumflex over (P)}(z) based on a manipulated variable u(k) supplied to the controlled object 103 at a timing k and an actual controlled variable y(k) to the manipulated variable u(k). Estimation of each parameter is repeatedly performed until a difference {tilde over (e)}(k) between the output from the model identifier 111 to the given manipulated variable u(k) (that is, the controlled variable ŷ(k) of the identification model {circumflex over (P)}(z) to the manipulated variable u(k)) and the actual control variable y(k) reaches a predetermined value (near zero) or below.
Particular identification (estimation) calculation in the adaptive identifier 107 is performed using a known adaptive update rule for a discrete-time transfer function model which is explained below.
If the discrete-time transfer function model P(z) of the controlled object 103 is expressed by equation (1), the actual controlled variable y(k) at a timing k is defined by equation (2).
                                              ⁢                              P            ⁡                          (              z              )                                =                                                                      b                                      n                    +                    1                                                  ⁢                                  z                  n                                            +                                                b                  n                                ⁢                                  z                                      n                    -                    1                                                              +              …              +                                                b                  2                                ⁢                z                            +                              b                1                                                                    z                n                            +                                                a                  n                                ⁢                                  z                                      n                    -                    1                                                              +              …              +                                                a                  2                                ⁢                z                            +                              a                1                                                                        (        1        )                                                          ⁢                                            y              ⁡                              (                k                )                                      =                                          θ                T                            ⁢                              v                ⁡                                  (                  k                  )                                                              ⁢                                          ⁢                                          ⁢          where          ⁢                                          ⁢                      θ            =                          [                                                b                                      n                    +                    1                                                  ,                …                ⁢                                                                  ,                                  b                  1                                ,                                  a                  n                                ,                …                ⁢                                                                  ,                                  a                  1                                            ]                                ⁢                                          ⁢                                    v              ⁡                              (                k                )                                      =                          [                                                u                  ⁡                                      (                    k                    )                                                  ,                                  u                  ⁡                                      (                                          k                      -                      1                                        )                                                  ,                …                ⁢                                                                  ,                                  u                  ⁡                                      (                                          k                      -                      n                                        )                                                  ,                                  -                                      y                    ⁡                                          (                                              n                        -                        1                                            )                                                                      ,                …                ⁢                                                                  ,                                  -                                      y                    ⁡                                          (                                              k                        -                        n                                            )                                                                                  ]                                                          (        2        )            
Here, why y(k) is defined by equation (2) is particularly explained, taking up the case when n=4, for example. When n=4, equation (1) can be expressed by equation (3) below.
                                                                        P                ⁡                                  (                  z                  )                                            =                                                                                          b                      5                                        ⁢                                          z                      4                                                        +                                                            b                      4                                        ⁢                                          z                                                                                                                        ⁢                        3                                                                              +                                                            b                      3                                        ⁢                                          z                                                                                                                        ⁢                        2                                                                              +                                                            b                      2                                        ⁢                                          z                                                                                                                            +                                      b                    1                                                                                        z                    4                                    +                                                            a                      4                                        ⁢                                          z                                                                                                                        ⁢                        3                                                                              +                                                            a                      3                                        ⁢                                          z                                                                                                                        ⁢                        2                                                                              +                                                            a                      2                                        ⁢                                          z                                                                                                                            +                                      a                    1                                                                                                                          =                                                                    b                    5                                    +                                                            b                      4                                        ⁢                                          z                                              -                        1                                                                              +                                                            b                      3                                        ⁢                                          z                                                                                                                        ⁢                                                  -                          2                                                                                                      +                                                            b                      2                                        ⁢                                          z                                                                                                                        ⁢                                                  -                          3                                                                                                      +                                                            b                      1                                        ⁢                                          z                                              -                        4                                                                                                              1                  +                                                            a                      4                                        ⁢                                          z                                              -                        1                                                                              +                                                            a                      3                                        ⁢                                          z                                                                                                                        ⁢                                                  -                          2                                                                                                      +                                                            a                      2                                        ⁢                                          z                                                                                                                        ⁢                                                  -                          3                                                                                                      +                                                            a                      1                                        ⁢                                          z                                              -                        4                                                                                                                                                    (        3        )            
Also, equation (4) below is established between the manipulated variable u(k) and the controlled variable y(k) to the controlled object 103.
                              P          ⁡                      (            z            )                          =                              y            ⁡                          (              k              )                                            u            ⁡                          (              k              )                                                          (        4        )            
Accordingly, substituting equation (3) into equation (4) results in equation (5) below.(b5+b4z−1+b3z−2+b2z−3+b1z−4)u(k)=(1+a4z−1+a3z−2+a2z−3+a1z−4)y(k)  (5)
Here, ‘z’ is a known delay operator, and ‘z−n’ means to delay time (timing) by n timing. Accordingly, equation (5) is reexpressed to define y(k) by equation (6) below.
                                                                        y                ⁡                                  (                  k                  )                                            =                            ⁢                                                                    (                                                                  b                        5                                            +                                                                        b                          4                                                ⁢                                                  z                                                      -                            1                                                                                              +                                                                        b                          3                                                ⁢                                                  z                                                      -                            2                                                                                              +                                                                        b                          2                                                ⁢                                                  z                                                      -                            3                                                                                              +                                                                        b                          1                                                ⁢                                                  z                                                      -                            4                                                                                                                )                                    ⁢                                      u                    ⁡                                          (                      k                      )                                                                      -                                                                                                      ⁢                                                (                                                                                    a                        4                                            ⁢                                              z                                                  -                          1                                                                                      +                                                                  a                        3                                            ⁢                                              z                                                  -                          2                                                                                      +                                                                  a                        2                                            ⁢                                              z                                                  -                          3                                                                                      +                                                                  a                        1                                            ⁢                                              z                                                  -                          4                                                                                                      )                                ⁢                                  y                  ⁡                                      (                    k                    )                                                                                                                          =                            ⁢                                                                    b                    5                                    ⁢                                      u                    ⁡                                          (                      k                      )                                                                      +                                                      b                    4                                    ⁢                                      u                    ⁡                                          (                                              k                        -                        1                                            )                                                                      +                                                      b                    3                                    ⁢                                      u                    ⁡                                          (                                              k                        -                        2                                            )                                                                      +                                                      b                    2                                    ⁢                                      u                    ⁡                                          (                                              k                        -                        3                                            )                                                                      +                                                                                                      ⁢                                                                    b                    2                                    ⁢                                      u                    ⁡                                          (                                              k                        -                        3                                            )                                                                      +                                                      b                    1                                    ⁢                                      u                    ⁡                                          (                                              k                        -                        4                                            )                                                                      -                                  {                                                                                                                                                                        a                              4                                                        ⁢                                                          y                              ⁡                                                              (                                                                  k                                  -                                  1                                                                )                                                                                                              +                                                                                    a                              3                                                        ⁢                                                          y                              ⁡                                                              (                                                                  k                                  -                                  2                                                                )                                                                                                              +                                                                                                                                                                                                                        a                              2                                                        ⁢                                                          y                              ⁡                                                              (                                                                  k                                  -                                  3                                                                )                                                                                                              +                                                                                    a                              1                                                        ⁢                                                          y                              ⁡                                                              (                                                                  k                                  -                                  4                                                                )                                                                                                                                                                                          }                                                                                        (        6        )            
The equations (3) to (6) show the examples when n=4. Equation (6), if generalized, can be expressed as the aforementioned equation (2).
If each parameter to be estimated, that is, an identification parameter {circumflex over (θ)}(k) (an estimated value of θ) is defined by equation (7) below, estimate calculation of this identification parameter {circumflex over (θ)}(k) is carried out according to the adaptive update rule defined by equation (8) below. L in equation (8) indicates adaptive gain.{circumflex over (θ)}(k)=[{circumflex over (b)}n+1, . . . {circumflex over (b)}1,ân, . . . , â1]  (7){circumflex over (θ)}(k)={circumflex over (θ)}(k−1)−Lv(k){tilde over (e)}(k)  (8)
Calculation according to the adaptive update rule of equation (8) is repeatedly performed until the difference {tilde over (e)}(k) is equal to a predetermined value or below as noted above. In other words, calculation according to the above adaptive update rule is carried out until the previously estimated identification parameter {circumflex over (θ)}(k−1) is nearly equal (ideally, completely equal) to the currently estimated identification parameter {circumflex over (θ)}(k). In this manner, {circumflex over (θ)}(k) is converged to a true value.
In this manner, when identifying a transfer function model of a controlled object according to adaptive control, the transfer function model is defined by a discrete-time transfer function in consideration that the control system according to adaptive control is actually mounted on various apparatus and devices. Thereafter, each coefficient (parameter) of a numerator and a denominator of the transfer function is estimated.